Apex-target condition for a ballistic jet ($u_y=\sqrt{2gh},u_x=w\sqrt{g/(2h)}$)

For a jet arriving horizontally at a target displaced horizontally by w and upward by h, the target lies at the projectile apex. Vertical kinematics gives $u_y^2=2gh$ and flight time $u_y/g$; the horizontal travel then fixes the displayed $u_x$. The launch speed is $\sqrt{u_x^2+u_y^2}$ and $\tan \theta =2h/w$. The ideal ballistic approximation neglects air drag, breakup and internal jet stresses after launch.

Steady incompressible mass flow rate stays constant along a jet, while speed and cross-sectional area can change under gravity. At a horizontal apex the arriving speed is the initial horizontal component. The force on a target that stops this horizontal momentum is $\dot{m}{v}_{\mathrm{hit}}$, using the conserved nozzle mass flux. Substituting the nozzle area into $\rho A_{\mathrm{hit}}{v}_{\mathrm{hit}}^2$ without adjusting the area gives the wrong result.